A new method for verifying the hyperbolicity of finitely presented groups Derek Holt University of Warwick Stevens Institute, Conference on Groups and Computation, Paul Schupp Celebration, 30th June 2017 Derek Holt (University of Warwick) June, 2017 1 / 28
Contents 1 Group presentations 2 Programs to verify hyperbolicity 3 van Kampen diagrams and the Dehn function 4 Small cancellation 5 Curvature functions 6 The RSym curvature distribution scheme 7 Pregroups and coloured diagrams 8 Examples 9 Future plans Derek Holt (University of Warwick) June, 2017 2 / 28
Group presentations Throughout the talk G = � X | R � will be a group defined by a presentation with X and R finite. We put A := X ± 1 . So R is the set of defining relators of G , and R ⊆ A ∗ . We assume throughout that the words in R are cyclically reduced. Derek Holt (University of Warwick) June, 2017 3 / 28
Group presentations Throughout the talk G = � X | R � will be a group defined by a presentation with X and R finite. We put A := X ± 1 . So R is the set of defining relators of G , and R ⊆ A ∗ . We assume throughout that the words in R are cyclically reduced. We are interested in attempting to decide whether G is hyperbolic . It is known that there is no general algorithm for this purpose, but the hyperbolicity of G can be verified. Derek Holt (University of Warwick) June, 2017 3 / 28
Group presentations Throughout the talk G = � X | R � will be a group defined by a presentation with X and R finite. We put A := X ± 1 . So R is the set of defining relators of G , and R ⊆ A ∗ . We assume throughout that the words in R are cyclically reduced. We are interested in attempting to decide whether G is hyperbolic . It is known that there is no general algorithm for this purpose, but the hyperbolicity of G can be verified. There are many equivalent conditions for hyperbolicity of G . These include • Geodesic triangles in the Cayley graph Γ( G , A ) are δ -slim for some fixed δ > 0; • Two infinite geodesic paths in Γ( G , A ) from a common vertex either do not diverge, or they diverge exponentially; • The Dehn function ∆ G of G is linear; • G has a Dehn presentation (so the word-problem of G is solvable in linear time). Derek Holt (University of Warwick) June, 2017 3 / 28
Verifying hyperbolicity The programs in the KBMAG package (available as a standalone program or via GAP or Magma) can verify hyperbolicity. They do this by first finding a shortlex automatic structure for G and verifying that geodesic bigons in Γ( G , A ) are uniformly thin. A result of Papasoglu then enables us to conclude that the group is hyperbolic. Derek Holt (University of Warwick) June, 2017 4 / 28
Verifying hyperbolicity The programs in the KBMAG package (available as a standalone program or via GAP or Magma) can verify hyperbolicity. They do this by first finding a shortlex automatic structure for G and verifying that geodesic bigons in Γ( G , A ) are uniformly thin. A result of Papasoglu then enables us to conclude that the group is hyperbolic. The alternative methods we are discussing today are based on generalizations of small cancellation theory. The project was initiated by Richard Parker in about 2008. Significant contributions have been made by Roney-Dougal, Neunh¨ offer, Linton and others. Experimental programs have been written by Parker and Neunh¨ offer, there is a new implementation in GAP by Markus Pfeiffer, and a recent Magma implementation by the speaker. Derek Holt (University of Warwick) June, 2017 4 / 28
Advantages and disadvantages of KBMAG � In theory, given enough resources, KBMAG method can verify the hyperbolicity of any hyperbolic group G . It has been successful on difficult examples, such as the Fibonacci group F (2 , 9). and the Heineken group � x , y , z | [[ x , [ x , y ]] = z , [ y , [ y , z ]] = x , [ z , [ z , x ]] = y � . Derek Holt (University of Warwick) June, 2017 5 / 28
Advantages and disadvantages of KBMAG � In theory, given enough resources, KBMAG method can verify the hyperbolicity of any hyperbolic group G . It has been successful on difficult examples, such as the Fibonacci group F (2 , 9). and the Heineken group � x , y , z | [[ x , [ x , y ]] = z , [ y , [ y , z ]] = x , [ z , [ z , x ]] = y � . � It can calculate the growth series of G as a rational function. Derek Holt (University of Warwick) June, 2017 5 / 28
Advantages and disadvantages of KBMAG � In theory, given enough resources, KBMAG method can verify the hyperbolicity of any hyperbolic group G . It has been successful on difficult examples, such as the Fibonacci group F (2 , 9). and the Heineken group � x , y , z | [[ x , [ x , y ]] = z , [ y , [ y , z ]] = x , [ z , [ z , x ]] = y � . � It can calculate the growth series of G as a rational function. � It provides no reasonable estimate of the slimness constant δ or of the Dehn function of G . Derek Holt (University of Warwick) June, 2017 5 / 28
Advantages and disadvantages of KBMAG � In theory, given enough resources, KBMAG method can verify the hyperbolicity of any hyperbolic group G . It has been successful on difficult examples, such as the Fibonacci group F (2 , 9). and the Heineken group � x , y , z | [[ x , [ x , y ]] = z , [ y , [ y , z ]] = x , [ z , [ z , x ]] = y � . � It can calculate the growth series of G as a rational function. � It provides no reasonable estimate of the slimness constant δ or of the Dehn function of G . � It enables only a quadratic-time solution of the word problem. Derek Holt (University of Warwick) June, 2017 5 / 28
Advantages and disadvantages of KBMAG � In theory, given enough resources, KBMAG method can verify the hyperbolicity of any hyperbolic group G . It has been successful on difficult examples, such as the Fibonacci group F (2 , 9). and the Heineken group � x , y , z | [[ x , [ x , y ]] = z , [ y , [ y , z ]] = x , [ z , [ z , x ]] = y � . � It can calculate the growth series of G as a rational function. � It provides no reasonable estimate of the slimness constant δ or of the Dehn function of G . � It enables only a quadratic-time solution of the word problem. � It can only be applied to individual group presentations - not to infinite families. Derek Holt (University of Warwick) June, 2017 5 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. Derek Holt (University of Warwick) June, 2017 6 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. � They run in polynomial time (returning true or fail). In practice, when they work they do so much more quickly than KBMAG. Derek Holt (University of Warwick) June, 2017 6 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. � They run in polynomial time (returning true or fail). In practice, when they work they do so much more quickly than KBMAG. � Unlike KBMAG, they can be used on presentations with large numbers of generators or relators. Derek Holt (University of Warwick) June, 2017 6 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. � They run in polynomial time (returning true or fail). In practice, when they work they do so much more quickly than KBMAG. � Unlike KBMAG, they can be used on presentations with large numbers of generators or relators. � They can sometimes be applied by hand, and to infinite families of group presentations. Derek Holt (University of Warwick) June, 2017 6 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. � They run in polynomial time (returning true or fail). In practice, when they work they do so much more quickly than KBMAG. � Unlike KBMAG, they can be used on presentations with large numbers of generators or relators. � They can sometimes be applied by hand, and to infinite families of group presentations. � They provide a reasonable estimate of the Dehn function of G (which can be used to estimate the slimness constant). Derek Holt (University of Warwick) June, 2017 6 / 28
Disadvantages and advantages of the new methods � They are not guaranteed to succeed on all presentations of hyperbolic groups. KBMAG is more likely to succeed on short hard examples. � They run in polynomial time (returning true or fail). In practice, when they work they do so much more quickly than KBMAG. � Unlike KBMAG, they can be used on presentations with large numbers of generators or relators. � They can sometimes be applied by hand, and to infinite families of group presentations. � They provide a reasonable estimate of the Dehn function of G (which can be used to estimate the slimness constant). � If the presentation is itself a Dehn presentation, then the programs may be able to verify this property, and hence enable a fast linear-time solution of the word problem. Derek Holt (University of Warwick) June, 2017 6 / 28
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